Voigt profile

(Centered) Voigt
Probability density function Plot of the centered Voigt profile for four cases. Each case has a full width at half-maximum of very nearly 3.6. The black and red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively.
Cumulative distribution function
Parameters	$\gamma,\sigma>0$
Support	$x\in(-\infty,\infty)$
PDF	$\frac{\Re[w(z)]}{\sigma\sqrt{2\pi}}, ~~~z=\frac{x+i\gamma}{\sigma\sqrt{2}}$
CDF	(complicated - see text)
Mean	(not defined)
Median	$0$
Mode	$0$
Variance	(not defined)
Skewness	(not defined)
Ex. kurtosis	(not defined)
MGF	(not defined)
CF	$e^{-\gamma\|t\|-\sigma^2 t^2/2}$

In spectroscopy, the Voigt profile (named after Woldemar Voigt) is a line profile resulting from the convolution of two broadening mechanisms, one of which alone would produce a Gaussian profile (usually, as a result of the Doppler broadening), and the other would produce a Lorentzian profile. Voigt profiles are common in many branches of spectroscopy and diffraction. Due to the computational expense of the convolution operation, the Voigt profile is often approximated using a pseudo-Voigt profile.

All normalized line profiles can be considered to be probability distributions. The Gaussian profile has a Gaussian, or normal, distribution and a Lorentzian profile has a Lorentz, or Cauchy, distribution. Without loss of generality, we can consider only centered profiles, which peak at zero. The Voigt profile is then a convolution of a Lorentz profile and a Gaussian profile:

V(x;\sigma ,\gamma )=\int _{-\infty }^{\infty }G(x';\sigma )L(x-x';\gamma )\,dx',

where x is the shift from the line center, $G(x;\sigma)$ is the centered Gaussian profile:

G(x;\sigma )\equiv {\frac {e^{-x^{2}/(2\sigma ^{2})}}{\sigma {\sqrt {2\pi }}}},

and $L(x;\gamma)$ is the centered Lorentzian profile:

L(x;\gamma )\equiv {\frac {\gamma }{\pi (x^{2}+\gamma ^{2})}}.

The defining integral can be evaluated as:

V(x;\sigma ,\gamma )={\frac {{\textrm {Re}}[w(z)]}{\sigma {\sqrt {2\pi }}}},

where Re[w(z)] is the real part of the Faddeeva function evaluated for

z=\frac{x+i\gamma}{\sigma\sqrt{2}}.

Properties

The Voigt profile is normalized:

\int _{-\infty }^{\infty }V(x;\sigma ,\gamma )\,dx=1,

since it is a convolution of normalized profiles. The Lorentzian profile has no moments (other than the zeroth), and so the moment-generating function for the Cauchy distribution is not defined. It follows that the Voigt profile will not have a moment-generating function either, but the characteristic function for the Cauchy distribution is well defined, as is the characteristic function for the normal distribution. The characteristic function for the (centered) Voigt profile will then be the product of the two:

\varphi_f(t;\sigma,\gamma) = E(e^{ixt}) = e^{-\sigma^2t^2/2 - \gamma |t|}.

Since normal distributions and Cauchy distributions are stable distributions, they are each are closed under convolution (up to change of scale), and it follows that the Voigt distributions are also closed under convolution.

Cumulative distribution function

Using the above definition for z , the cumulative distribution function (CDF) can be found as follows:

F(x_{0};\mu ,\sigma )=\int _{-\infty }^{x_{0}}{\frac {\mathrm {Re} (w(z))}{\sigma {\sqrt {2\pi }}}}\,dx=\mathrm {Re} \left({\frac {1}{\sqrt {\pi }}}\int _{z(-\infty )}^{z(x_{0})}w(z)\,dz\right).

Substituting the definition of the Faddeeva function (scaled complex error function) yields for the indefinite integral:

{\frac {1}{\sqrt {\pi }}}\int w(z)\,dz={\frac {1}{\sqrt {\pi }}}\int e^{-z^{2}}\left[1-\mathrm {erf} (-iz)\right]\,dz,

which may be solved to yield

{\frac {1}{\sqrt {\pi }}}\int w(z)\,dz={\frac {\mathrm {erf} (z)}{2}}+{\frac {iz^{2}}{\pi }}\,_{2}F_{2}\left(1,1;{\frac {3}{2}},2;-z^{2}\right),

where ${}_{2}F_{2}()$ is a hypergeometric function. In order for the function to approach zero as x approaches negative infinity (as the CDF must do), an integration constant of 1/2 must be added. This gives for the CDF of Voigt:

F(x;\mu ,\sigma )=\mathrm {Re} \left[{\frac {1}{2}}+{\frac {\mathrm {erf} (z)}{2}}+{\frac {iz^{2}}{\pi }}\,_{2}F_{2}\left(1,1;{\frac {3}{2}},2;-z^{2}\right)\right].

The width of the Voigt profile

The full width at half maximum (FWHM) of the Voigt profile can be found from the widths of the associated Gaussian and Lorentzian widths. The FWHM of the Gaussian profile is

f_{\mathrm {G} }=2\sigma {\sqrt {2\ln(2)}}.

The FWHM of the Lorentzian profile is

f_{\mathrm {L} }=2\gamma .

Define Φ = $f_L/f_G$ . Then the FWHM of the Voigt profile ( $f_V$ ) can be estimated as

f_{\mathrm {V} }\approx f_{\mathrm {G} }\left(1-c_{0}c_{1}+{\sqrt {\phi ^{2}+2c_{1}\phi +c_{0}^{2}c_{1}^{2}}}\right),

where $c_{0}$ = 2.0056 and $c_{1}$ = 1.0593. This estimate has a standard deviation of error of about 2.4% for values of φ between 0 and 10. Note that the above equation is exactly correct in the limit of φ = 0 and φ = ∞, that is for pure Gaussian and Lorentzian profiles.

A better approximation with an accuracy of 0.02% is given by^[1]

f_\mathrm{V}\approx 0.5346 f_\mathrm{L}+\sqrt{0.2166f_\mathrm{L}^2+f_\mathrm{G}^2}.

This approximation is exactly correct for a pure Gaussian, but has an error of about 0.000305% for a pure Lorentzian profile.

The uncentered Voigt profile

If the Gaussian profile is centered at $\mu_G$ and the Lorentzian profile is centered at $\mu_L$ , the convolution is centered at $\mu_G+\mu_L$ and the characteristic function is

\varphi_f(t;\sigma,\gamma,\mu_\mathrm{G},\mu_\mathrm{L})= e^{i(\mu_\mathrm{G}+\mu_\mathrm{L})t-\sigma^2t^2/2 - \gamma |t|}.

The mode and median are both located at $\mu_G+\mu_L$ .

Voigt functions

The Voigt functions^[2] U, V, and H (sometimes called the line broadening function) are defined by

U(x,t)+iV(x,t)={\sqrt {\frac {\pi }{4t}}}e^{z^{2}}{\text{erfc}}(z)={\sqrt {\frac {\pi }{4t}}}w(iz),

H(a,u)={\frac {U(u/a,1/4a^{2})}{a{\sqrt {\pi }}}},

where

z=(1-ix)/2\sqrt t,

erfc is the complementary error function, and w(z) is the Faddeeva function.

Relation to Voigt profile

V(x;\sigma ,\gamma )=H(a,u)/({\sqrt {2}}{\sqrt {\pi }}\sigma ),

with

a=\gamma /({\sqrt {2}}\sigma )

and

u=x/({\sqrt {2}}\sigma ).

Pseudo-Voigt approximation

The pseudo-Voigt profile (or pseudo-Voigt function) is an approximation of the Voigt profile V(x) using a linear combination of a Gaussian curve G(x) and a Lorentzian curve L(x) instead of their convolution.

The pseudo-Voigt function is often used for calculations of experimental spectral line shapes.

The mathematical definition of the normalized pseudo-Voigt profile is given by

V_{p}(x)=\eta \cdot L(x,f)+(1-\eta )\cdot G(x,f)

with

0<\eta <1

There are several possible choices for the $\eta$ parameter.^[3]^[4]^[5]^[6] A simple formula, accurate to 1%, is^[7]^[8]

\eta =1.36603(f_{L}/f)-0.47719(f_{L}/f)^{2}+0.11116(f_{L}/f)^{3},

where

f=[f_{G}^{5}+2.69269f_{G}^{4}f_{L}+2.42843f_{G}^{3}f_{L}^{2}+4.47163f_{G}^{2}f_{L}^{3}+0.07842f_{G}f_{L}^{4}+f_{L}^{5}]^{1/5}.

References

↑ Olivero, J. J.; R. L. Longbothum (February 1977). "Empirical fits to the Voigt line width: A brief review". Journal of Quantitative Spectroscopy and Radiative Transfer. 17 (2): 233–236. Bibcode:1977JQSRT..17..233O. doi:10.1016/0022-4073(77)90161-3. ISSN 0022-4073. Retrieved 2009-04-01.
↑ Temme, N. M. (2010), "Voigt function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
↑ Wertheim, G. K. and Butler, M. A. and West, K. W. and Buchanan, D. N. E. (1974). "Determination of the Gaussian and Lorentzian content of experimental line shapes". Review of Scientific Instruments. 45 (11): 1369–1371. doi:10.1063/1.1686503.
↑ Sánchez-Bajo, F.; F. L. Cumbrera (August 1997). "The Use of the Pseudo-Voigt Function in the Variance Method of X-ray Line-Broadening Analysis". Journal of Applied Crystallography. 30 (4): 427–430. doi:10.1107/S0021889896015464. Retrieved 2014-07-31.
↑ Liu, Yuyan and Lin, Jieli and Huang, Guangming and Guo, Yuanqing and Duan, Chuanxi (2001). "Simple empirical analytical approximation to the Voigt profile". JOSA B. 18 (5): 666–672. doi:10.1364/josab.18.000666.
↑ Di Rocco, HO & Cruzado, A (2012). "The Voigt Profile as a Sum of a Gaussian and a Lorentzian Functions, when the Weight Coefficient Depends Only on the Widths Ratio". Acta Physica Polonica A. 122 (4): 666–669.
↑ Ida, T and Ando, M and Toraya, H (2000). "Extended pseudo-Voigt function for approximating the Voigt profile". Journal of Applied Crystallography. 33 (6): 1311–1316. doi:10.1107/s0021889800010219.
↑ P. Thompson, D. E. Cox and J. B. Hastings (1987). "Rietveld refinement of Debye-Scherrer synchrotron X-ray data from Al₂O₃". Journal of Applied Crystallography. 20: 79–83. doi:10.1107/S0021889887087090.

External links

http://apps.jcns.fz-juelich.de/libcerf, numeric C library for complex error functions, provides a function voigt(x, sigma, gamma) with approximately 13–14 digits precision.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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